Structure for Regular Inclusions. II: Cartan envelopes, pseudo-expectations and twists

TL;DR

This paper develops the theory of Cartan envelopes for regular inclusions, characterizing their existence via pseudo-expectations and constructing associated twisted groupoids, advancing the understanding of the structure of such inclusions.

Contribution

It introduces the concept of Cartan envelopes for regular inclusions, providing existence criteria, characterizations, and explicit constructions using twists and groupoids.

Findings

Existence of Cartan envelopes is equivalent to the unique faithful pseudo-expectation property.

Constructs Hausdorff twisted groupoids for covering inclusions.

Characterizes Cartan envelopes in terms of ideal intersection and states on C.

Abstract

We introduce the notion of a Cartan envelope for a regular inclusion (C,D). When a Cartan envelope exists, it is the unique, minimal Cartan pair into which (C,D) regularly embeds. We prove a Cartan envelope exists if and only if (C,D) has the unique faithful pseudo-expectation property and also give a characterization of the Cartan envelope using the ideal intersection property. For any covering inclusion, we construct a Hausdorff twisted groupoid using appropriate linear functionals and we give a description of the Cartan envelope for (C,D) in terms of a twist whose unit space is a set of states on C constructed using the unique pseudo-expectation. For a regular MASA inclusion, this twist differs from the Weyl twist; in this setting, we show that the Weyl twist is Hausdorff precisely when there exists a conditional expectation of C onto D. We show that a regular inclusion with the unique pseudo-expectation property is a covering inclusion and give other consequences of the unique pseudo-expectation property.